boost::any has a perfect forward constructor declared as:

    template<typename ValueType>
    any(ValueType&& value
        , typename boost::disable_if<boost::is_same<any&, ValueType> >::type* = 0 // disable if value has type `any&`
        , typename boost::disable_if<boost::is_const<ValueType> >::type* = 0) // disable if value has type `const ValueType&&`
      : content(new holder< typename decay<ValueType>::type >(static_cast<ValueType&&>(value)))

The is_const SFINAE exclusion forces const types to the regular copy constructor:
    template<typename ValueType>
    any(const ValueType & value)
      : content(new holder<
            BOOST_DEDUCED_TYPENAME remove_cv<BOOST_DEDUCED_TYPENAME decay<const ValueType>::type>::type

What is different about how the regular copy constructor treats a constant value than what the perfect forward construct would do if the is_const exclusion were removed?